Overview

We will focus on the fundamentals behind observational studies this week.

Two common types

Sampling designs

What is “random” sampling?

We know that…

From the previous lecture

• Sample mean is a good measure of central tendency.
• Sample variance is a good measure of dispersion.
• Sample size affects the precision of the sample mean.

Combining an estimate with its precision

A confidence interval (CI) is:

• A range of values, derived from a sample of data, that is used to estimate the range of values for a population parameter
• Crucial for hypothesis testing and estimation, the basis of statistical inference

You will often see CIs in scientific papers, reports, and news articles.

Calculating confidence intervals

What we need

1. Estimate of the population parameter, i.e. the sample mean (\bar{x})
2. Critical value (t_{n-1}) from the sampling distribution of the estimate, which depends on the number of samples and the confidence level. This is usually based on the t-distribution
3. Standard error of the estimate, standardised by the number of samples i.e. SE of the mean (SE_{\bar{x}})

All these (mean, standard error and critical value) are combined to form the confidence interval.

Breakdown

In general, a CI has the form: \text{estimate} \pm \text{margin of error}

where the margin of error is a function of the standard error of the estimate:

\text{estimate} \pm (\text{critical value} \times \text{standard error (estimate)})

where the critical value is based on the sampling distribution of the estimate i.e. the t-distribution.

Formula for 95% Confidence Interval (CI)

\bar{x} \pm \left(t_{n-1} \times \frac{s}{\sqrt{n}}\right)

Step-by-step calculation by hand

1. Calculate the sample mean, \bar{x}

2. Calculate the sample standard deviation, s

3. Determine the standard error of the mean, SE_{\bar{x}} = \frac{s}{\sqrt{n}}

4. Look up the t-value, t_{n-1}, from the t-distribution table for the 95% confidence level and n-1 degrees of freedom (manual table or function in R).

5. Compute the margin of error: \text{Margin of Error} = t_{n-1} \times SE_{\bar{x}}

6. Finally, the 95% CI is: \bar{x} \pm (t_{n-1} \times SE_{\bar{x}})

You need to be able to calculate this by hand/calculator.

More definitions

Degrees of freedom (df)

The number of independent values in a sample. It is calculated as: \text{df} = n - 1 where n is the number of samples. We subtract 1 because of Bessel’s correction.

More definitions

t-critical value

Given a confidence level (e.g. 95%) and degrees of freedom (df), the t-critical value is a value that is used to determine the margin of error in a confidence interval.

It assumes: - A t-distribution of the data - A symmetric distribution around the mean

More definitions

t-distribution

A probability distribution that is used to estimate the population. Using the correct distribution gives better estimates.

• Similar to the normal distribution, but with heavier tails.
• As the sample size increases, the t-distribution approaches the normal distribution, so we do not need to worry about it for large sample sizes.

library(ggplot2)
library(gganimate)

# Variable to control animation speed (higher values = slower transitions)
anim_speed <- 1

# Create a sequence of x values for the plot
x <- seq(-4, 4, length.out = 400)

# Degrees of freedom: 1 through 5 then even numbers from 6 to 30
dfs <- c(1:5, seq(6, 30, by = 2))

# Prepare data for t-distribution curves with a new "df" column
t_curves <- do.call(rbind, lapply(dfs, function(df) {
  data.frame(x = x, 
             density = dt(x, df), 
             df = df)
}))

# Create data for the standard normal distribution (static)
normal_curve <- data.frame(x = x, 
                             density = dnorm(x))

# Plot with gganimate: animate t-distribution curves (each frame shows one df)
p <- ggplot() +
  # Static normal distribution curve
  geom_line(data = normal_curve, aes(x = x, y = density), 
            color = "black", linetype = "dashed", size = 1) +
  # t-distribution curve that will animate
  geom_line(data = t_curves, aes(x = x, y = density, color = factor(df)), 
            size = 1) +
  labs(title = "Degrees of Freedom: {closest_state}",
       x = "x",
       y = "Density") +
  theme(legend.position = "none") +
  transition_states(states = df, transition_length = anim_speed, state_length = anim_speed)

p

Interpreting confidence intervals

• Confidence intervals depend on a specified confidence level (e.g. 95%, 99%) with higher confidence levels producing wider intervals (i.e. more conservative).
• Another way to think of it: a range of values that we are fairly sure contains the true value of the population parameter.

Soil carbon

Soil carbon content was measured at 7 locations across the area. The amount at each location was 48, 56, 90, 78, 86, 71, 42 tonnes per hectare (t/ha).

soil <- c(48, 56, 90, 78, 86, 71, 42)
soil

[1] 48 56 90 78 86 71 42

What is the mean soil carbon content and how confident are we in this estimate? How this is calculated depends on whether we used simple random sampling or stratified random sampling.

Mean and 95% CI

Step-by-step calculation

Implementation in R

Manual calculation

# Step 1: Calculate the sample mean of soil carbon content
mean_soil <- mean(soil)

# Step 2: Calculate the sample standard deviation of soil carbon content
sd_soil <- sd(soil)

# Step 3: Calculate the standard error of the mean (SE)
se_soil <- sd_soil / sqrt(length(soil))

# Step 4: Calculate the t-critical value for a 95% confidence interval
t_crit <- qt(0.975, df = length(soil) - 1)

# Step 5: Calculate the margin of error (t_crit * SE)
# and then determine the lower and upper bounds of the confidence interval
ci <- mean_soil + c(-1, 1) * (t_crit * se_soil)

# Step 6: View the calculated 95% confidence interval
ci

[1] 49.84627 84.72516

There are ways to calculate this in R quickly, but it is important to understand the manual calculation.

Questions

• How precise is our estimate?
• How big a change must there be to estimate a statistically significant change?
• Can we sample more efficiently?

To answer these questions, we need to compare simple random sampling with a hypothetical stratified random sampling design (i.e. what if we had considered stratification before sampling?)